Third-order derivative matrix of a skew ray with respect to the source ray vector at a flat boundary

Our group recently showed that the Seidel primary ray aberration coefficients of an axis-symmetrical system can be accurately determined using the third-order Taylor series expansion of a skew ray R^¯_m on an image plane. This finding inspires us to determine the third-order derivative matrix of R^¯_m with respect to the vector X^¯₀ of the source ray, i.e., ∂R^{¯ 3}_m/∂X^{¯ 3}₀, under reflection/refraction at a flat boundary. Finite difference methods using the second-order derivative matrix, ∂R^{¯ 2}_m/∂X^{¯ 2}₀, require multiple rays to compute ∂R^{¯ 3}_m/∂X^{¯ 3}₀ and suffer from cumulative rounding and truncation errors. By contrast, the present method is based on differential geometry. Thus, it provides a greater inherent accuracy and requires the tracing of just one ray. The proposed method facilitates the analytical investigation of the primary aberrations of an axis-symmetrical system and can be easily extended to determine the higher-order derivative matrices required to explore higher-order ray aberration coefficients.